Maharashtra State BoardHSC Science (General) 11th
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Answer the following question. Derive equations of motion for a particle moving in a plane and show that the motion can be resolved in two independent motions in mutually perpendicular directions. - Physics

Answer in Brief

Answer the following question.

Derive equations of motion for a particle moving in a plane and show that the motion can be resolved in two independent motions in mutually perpendicular directions.

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Solution

  1. Consider an object moving in an x-y plane. Let the initial velocity of the object be `vec"u"` at t = 0 and its velocity at time t be `vec"v"`
  2. As the acceleration is constant, the average acceleration and the instantaneous acceleration will be equal.
    `vec"a"_"av" = (vec"v"_2 - vec"v"_1)/("t"_2 - "t"_1) = (("v"_"2x" - "v"_"1x")/("t"_2 - "t"_1))hat"i" + (("v"_"2y" - "v"_"1y")/("t"_2 - "t"_1))`
    ∴ `vec"a" = ((vec"v" - vec"u"))/(("t - 0"))`
    ∴ `vec"v" = vec"u" + vec"a""t"`      ....(1)
    This is the first equation of motion in vector form.
  3. Let the displacement of the object from time t = 0 to t be `vec "s"`
    For constant acceleration, `vec"v"_"av" = (vec"v" + vec"u")/2`
    `vec"s" = (vec"v"_"av")"t" = ((vec"v" + vec"u")/2)"t" = ((vec"u" + vec"u" + vec"a""t")/2)"t"`
    ∴ `vec"s" = vec"u""t" + 1/2 vec"a""t"^2`   ...(2)
    This is the second equation of motion in vector form.
  4. Equations (1) and (2) can be resolved into their x and y components so as to get corresponding scalar equations as follows.
    `"v"_"x" = "u"_"x" + "a"_"x" "t"`    ....(3)
    `"v"_"y" = "u"_"y" + "a"_"y" "t"`    ....(4)
    `"s"_"x" = "u"_"x" "t" + 1/2 "a"_"x""t"^2`   ...(5)
    `"s"_"y" = "u"_"y""t" + 1/2"a"_"y""t"^2`   ....(6)
  5. It can be seen that equations (3) and (5) involve only the x components of displacement, velocity and acceleration while equations (4) and (6) involve only the y components of these quantities.
  6. Thus, the motion along the x-direction of the object is completely controlled by the x components of velocity and acceleration while that along the y-direction is completely controlled by the y components of these quantities.
  7. This shows that the two sets of equations are independent of each other and can be solved independently.
  Is there an error in this question or solution?
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APPEARS IN

Balbharati Physics 11th Standard Maharashtra State Board
Chapter 3 Motion in a Plane
Exercises | Q 2. (v) | Page 45
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